Introduction to Quantum Mechanics · Problem 2.27

Time-Independent Schrödinger Equation — David J. Griffiths

Physics — Quantum Mechanics · double_delta_function

Problem

Consider the double delta-function potential

V(x) = -\alpha[\delta(x+a) + \delta(x-a)],

where

\alpha

and

a

are positive constants. (a) Sketch this potential. (b) How many bound states does it possess? Find the allowed energies, for

\alpha = \hbar^2/ma

and for

\alpha = \hbar^2/4ma

, and sketch the wave functions. (c) What are the bound state energies in the limiting cases (i)

a \to 0

and (ii)

a \to \infty

(holding

\alpha

fixed)? Explain why your answers are reasonable, by comparison with the single delta-function well.